Which is the most important leg of a three-legged stool?

In The Republic, Plato writes about the human soul. It has a contemplative or thinking part, a part for carrying out the actions of its thinking, and a part for higher reasoning. I believe these are very roughly analogous to the branches of the US government. The legislative branch would be the contemplative or thinking branch. The executive branch carries out the decisions of both the other two branches, and the judicial branch is for higher reasoning.

(Philosophers and those who’ve read The Republic since I have which was 15 years ago, please, correct me.)

Important and powerful are 2 separate words. They are not synonymous. This is really 2 questions, which is sneaky.

In my opinion, the soul requires all 3 parts to function properly, and the government requires all 3 branches to work effectively. There is no single portion or branch that is more important or powerful than the other two.

Damn, now I’ve got another book on my reading list.

Executive has the most power concentration, followed by the SCOTUS, followed by Congress. They are the chisel, the crowbar, and the hammer of the American Federal toolbox.
IIRC the POTUS is the only one with an established line of succession, so while not the most important branch, he is the most easily pruned and so people worry more about him.

Important? Mu; that question is based on false presumptions and therefore cannot be answered.

Powerful? Due to checks and balances, the Executive can (in some cases) override the Legislative and totally do away with the Judicial. Also, which branch controls the Military? While Congress controls the purse strings, that control only really works when you don’t get mugged. And Martial Law kind of sidesteps the courts.

Most powerful: Currently (and sadly) the Executive.

Most important: they’re all supposed to be equally important (and powerful)

They’re all theoretically important, if you look at the Constitutional foundation for all three, and their individual and collective influence on American history. But as far as power, these days it’s probably a toss-up between the Executive and the Judicial. The Legislative branch was more or less emasculated a while ago, and with the present gridlock, rendering itself irrelevant, bit by bit.